12. List the all properties of rational numbers.
Answers
ᴛʜᴇ ᴍᴀᴊᴏʀ ᴘʀᴏᴘᴇʀᴛɪᴇs ᴏғ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs ᴀʀᴇ:
ᴄʟᴏsᴜʀᴇ ᴘʀᴏᴘᴇʀᴛʏ
ᴄᴏᴍᴍᴜᴛᴀᴛɪᴠɪᴛʏ ᴘʀᴏᴘᴇʀᴛʏ
ᴀssᴏᴄɪᴀᴛɪᴠᴇ ᴘʀᴏᴘᴇʀᴛʏ
ᴅɪsᴛʀɪʙᴜᴛɪᴠᴇ ᴘʀᴏᴘᴇʀᴛʏ
ʟᴇᴛ ᴜs ɴᴏᴡ sᴛᴜᴅʏ ᴛʜᴇsᴇ ᴘʀᴏᴘᴇʀᴛɪᴇs ɪɴ ᴅᴇᴛᴀɪʟ.
ᴄʟᴏsᴜʀᴇ ᴘʀᴏᴘᴇʀᴛʏ

sᴏᴜʀᴄᴇ: sᴏʟᴠɪɴɢ ᴍᴀᴛʜ ᴘʀᴏʙʟᴇᴍs
1) ᴀᴅᴅɪᴛɪᴏɴ ᴏғ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs
ᴛʜᴇ ᴄʟᴏsᴜʀᴇ ᴘʀᴏᴘᴇʀᴛʏ sᴛᴀᴛᴇs ᴛʜᴀᴛ ғᴏʀ ᴀɴʏ ᴛᴡᴏ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs ᴀ ᴀɴᴅ ʙ, ᴀ + ʙ ɪs ᴀʟsᴏ ᴀ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ.
12 + 34
= 4+68
= 108
ᴏʀ, = 54
ᴛʜᴇ ʀᴇsᴜʟᴛ ɪs ᴀ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ. sᴏ ᴡᴇ sᴀʏ ᴛʜᴀᴛ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs ᴀʀᴇ ᴄʟᴏsᴇᴅ ᴜɴᴅᴇʀ ᴀᴅᴅɪᴛɪᴏɴ.
2) sᴜʙᴛʀᴀᴄᴛɪᴏɴ ᴏғ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs
ᴛʜᴇ ᴄʟᴏsᴜʀᴇ ᴘʀᴏᴘᴇʀᴛʏ sᴛᴀᴛᴇs ᴛʜᴀᴛ ғᴏʀ ᴀɴʏ ᴛᴡᴏ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs ᴀ ᴀɴᴅ ʙ, ᴀ – ʙ ɪs ᴀʟsᴏ ᴀ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ.
12 – 34
= 4–68
= −28
ᴏʀ, = −14
ᴛʜᴇ ʀᴇsᴜʟᴛ ɪs ᴀ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ. sᴏ ᴛʜᴇ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs ᴀʀᴇ ᴄʟᴏsᴇᴅ ᴜɴᴅᴇʀ sᴜʙᴛʀᴀᴄᴛɪᴏɴ.
3) ᴍᴜʟᴛɪᴘʟɪᴄᴀᴛɪᴏɴ ᴏғ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs
ᴛʜᴇ ᴄʟᴏsᴜʀᴇ ᴘʀᴏᴘᴇʀᴛʏ sᴛᴀᴛᴇs ᴛʜᴀᴛ ғᴏʀ ᴀɴʏ ᴛᴡᴏ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs ᴀ ᴀɴᴅ ʙ, ᴀ × ʙ ɪs ᴀʟsᴏ ᴀ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ.
ɴᴜᴍʙᴇʀ.
12 × 34
= 68
ᴛʜᴇ ʀᴇsᴜʟᴛ ɪs ᴀ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ. sᴏ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs ᴀʀᴇ ᴄʟᴏsᴇᴅ ᴜɴᴅᴇʀ ᴍᴜʟᴛɪᴘʟɪᴄᴀᴛɪᴏɴ.
4) ᴅɪᴠɪsɪᴏɴ ᴏғ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs
ᴛʜᴇ ᴄʟᴏsᴜʀᴇ ᴘʀᴏᴘᴇʀᴛʏ sᴛᴀᴛᴇs ᴛʜᴀᴛ ғᴏʀ ᴀɴʏ ᴛᴡᴏ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs ᴀ ᴀɴᴅ ʙ, ᴀ ÷ ʙ ɪs ᴀʟsᴏ ᴀ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ.
12 ÷ 34
= 1×42×3
= 23
ᴛʜᴇ ʀᴇsᴜʟᴛ ɪs ᴀ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ. ʙᴜᴛ ᴡᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ ᴀɴʏ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ ᴀ, ᴀ ÷ 0 ɪs ɴᴏᴛ ᴅᴇғɪɴᴇᴅ. sᴏ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs ᴀʀᴇ ɴᴏᴛ ᴄʟᴏsᴇᴅ ᴜɴᴅᴇʀ ᴅɪᴠɪsɪᴏɴ. ʙᴜᴛ ɪғ ᴡᴇ ᴇxᴄʟᴜᴅᴇ 0, ᴛʜᴇɴ ᴀʟʟ ᴛʜᴇ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs ᴀʀᴇ ᴄʟᴏsᴇᴅ ᴜɴᴅᴇʀ ᴅɪᴠɪsɪᴏɴ.
ᴄᴏᴍᴍᴜᴛᴀᴛɪᴠᴇ ᴘʀᴏᴘᴇʀᴛʏ
1. ᴀᴅᴅɪᴛɪᴏɴ
ғᴏʀ ᴀɴʏ ᴛᴡᴏ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs ᴀ ᴀɴᴅ ʙ, ᴀ + ʙ = ʙ+ ᴀ
−23+ 57 ᴀɴᴅ 57+ −23 = 121
sᴏ, −23+ 57 = 57+ −23
ᴡᴇ sᴇᴇ ᴛʜᴀᴛ ᴛʜᴇ ᴛᴡᴏ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs ᴄᴀɴ ʙᴇ ᴀᴅᴅᴇᴅ ɪɴ ᴀɴʏ ᴏʀᴅᴇʀ. sᴏ ᴀᴅᴅɪᴛɪᴏɴ ɪs ᴄᴏᴍᴍᴜᴛᴀᴛɪᴠᴇ ғᴏʀ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs.
2. sᴜʙᴛʀᴀᴄᴛɪᴏɴ
ғᴏʀ ᴀɴʏ ᴛᴡᴏ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs ᴀ ᴀɴᴅ ʙ, ᴀ – ʙ ≠ ʙ – ᴀ. ɢɪᴠᴇɴ ᴀʀᴇ ᴛʜᴇ ᴛᴡᴏ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs 53 ᴀɴᴅ 14,
53 – 14 = 20−312
= 1712
ʙᴜᴛ, 14 – 53 = 3−2012
= −1712
sᴏ sᴜʙᴛʀᴀᴄᴛɪᴏɴ ɪs ɴᴏᴛ ᴄᴏᴍᴍᴜᴛᴀᴛɪᴠᴇ ғᴏʀ ʀᴀᴛɪᴏᴀɴʟ ɴᴜᴍʙᴇʀs.
3. ᴍᴜʟᴛɪᴘʟɪᴄᴀᴛɪᴏɴ
ғᴏʀ ᴀɴʏ ᴛᴡᴏ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs ᴀ ᴀɴᴅ ʙ, ᴀ × ʙ = ʙ × ᴀ
−73+ 65 = 65+ −73
= −4215 = −4215
ᴡᴇ sᴇᴇ ᴛʜᴀᴛ ᴛʜᴇ ᴛᴡᴏ ʀᴀᴛʀɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs ᴄᴀɴ ʙᴇ ᴍᴜʟᴛɪᴘʟɪᴇᴅ ɪɴ ᴀɴʏ ᴏʀᴅᴇʀ. sᴏ ᴍᴜʟᴛɪᴘʟɪᴄᴀᴛɪᴏɴ ɪs ᴄᴏᴍᴍᴜᴛᴀᴛɪᴠᴇ ғᴏʀ ʀᴀᴛɪᴏᴀɴʟ ɴᴜᴍʙᴇʀs.
4. ᴅɪᴠɪsɪᴏɴ
ғᴏʀ ᴀɴʏ ᴛᴡᴏ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs ᴀ ᴀɴᴅ ʙ, ᴀ ÷ ʙ ≠ ʙ ÷ ᴀ. ɢɪᴠᴇɴ ᴀʀᴇ ᴛʜᴇ ᴛᴡᴏ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs 53ᴀɴᴅ 14
53 ÷ 14 = 5×43×1
= 203
ʙᴜᴛ, 14 ÷ 53 = 1×34×5
= 320
ᴡᴇ sᴇᴇ ᴛʜᴀᴛ ᴛʜᴇ ᴇxᴘʀᴇssɪᴏɴs ᴏɴ ʙᴏᴛʜ ᴛʜᴇ sɪᴅᴇs ᴀʀᴇ ɴᴏᴛ ᴇϙᴜᴀʟ. sᴏ ᴅɪᴠsɪᴏɴ ɪs ɴᴏᴛ ᴄᴏᴍᴍᴜᴛᴀᴛɪᴠᴇ ғᴏʀ ʀᴀᴛɪᴏᴀɴᴀʟ ɴᴜᴍʙᴇʀs.
ᴀssᴏᴄɪᴀᴛɪᴠᴇ ᴘʀᴏᴘᴇʀᴛʏ
ᴛᴀᴋᴇ ᴀɴʏ ᴛʜʀᴇᴇ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs ᴀ, ʙ ᴀɴᴅ ᴄ. ғɪʀsᴛʟʏ ᴀᴅᴅ ᴀ ᴀɴᴅ ʙ ᴀɴᴅ ᴛʜᴇɴ ᴀᴅᴅ ᴄ ᴛᴏ ᴛʜᴇ sᴜᴍ. (ᴀ + ʙ) + ᴄ. ɴᴏᴡ ᴀɢᴀɪɴ ᴀᴅᴅ ʙ ᴀɴᴅ ᴄ ᴀɴᴅ ᴛʜᴇɴ ᴀ ᴛᴏ ᴛʜᴇ sᴜᴍ, ᴀ + (ʙ + ᴄ). ɪs (ᴀ + ʙ) + ᴄ ᴀɴᴅ ᴀ + (ʙ + ᴄ) sᴀᴍᴇ? ʏᴇs ᴀɴᴅ ᴛʜɪs ɪs ʜᴏᴡ ᴀssᴏᴄɪᴀᴛɪᴠᴇ ᴘʀᴏᴘᴇʀᴛʏ ᴡᴏʀᴋs. ɪᴛ sᴛᴀᴛᴇs ᴛʜᴀᴛ ʏᴏᴜ ᴄᴀɴ ᴀᴅᴅ ᴏʀ ᴍᴜʟᴛɪᴘʟʏ ɴᴜᴍʙᴇʀs ʀᴇɢᴀʀᴅʟᴇss ᴏғ ʜᴏᴡ ᴛʜᴇʏ ᴀʀᴇ ɢʀᴏᴜᴘᴇᴅ.
ғᴏʀ ᴇxᴀᴍᴘʟᴇ, ɢɪᴠᴇɴ ɴᴜᴍʙᴇʀs ᴀʀᴇ 5, -6 ᴀɴᴅ 23
( 5 – 6 ) + 23
= -1 + 23
= – 13
ɴᴏᴡ, 5 + ( -6 + 23 )
= – 13
ɪɴ ʙᴏᴛʜ ᴛʜᴇ ɢʀᴏᴜᴘs ᴛʜᴇ sᴜᴍ ɪs ᴛʜᴇ sᴀᴍᴇ.
ᴀᴅᴅɪᴛɪᴏɴ ᴀɴᴅ ᴍᴜʟᴛɪᴘʟɪᴄᴀᴛɪᴏɴ ᴀʀᴇ ᴀssᴏᴄɪᴀᴛɪᴠᴇ ғᴏʀ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs.
sᴜʙᴛʀᴀᴄᴛɪᴏɴ ᴀɴᴅ ᴅɪᴠɪsɪᴏɴ ᴀʀᴇ ɴᴏᴛ ᴀssᴏᴄɪᴀᴛɪᴠᴇ ғᴏʀ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs.
ᴅɪsᴛʀɪʙᴜᴛɪᴠᴇ ᴘʀᴏᴘᴇʀᴛʏ
ᴅɪsᴛʀɪʙᴜᴛɪᴠᴇ ᴘʀᴏᴘᴇʀᴛʏ sᴛᴀᴛᴇs ᴛʜᴀᴛ ғᴏʀ ᴀɴʏ ᴛʜʀᴇᴇ ɴᴜᴍʙᴇʀs x, ʏ ᴀɴᴅ ᴢ ᴡᴇ ʜᴀᴠᴇ
x × ( ʏ + ᴢ ) = (x × ʏ) +( x × ᴢ)